limits of sequences of functions and uniform convergence

Question

I have a sequence of functions $(f_n)$ defined by

$$ f_n(x) = \begin{cases} 0, & x=0,\\ x, & 0< x<1/n, \\ x^2, & 1/n <x. \end{cases} $$ I need to determine the limit for the function and to determine if $(f_n)$ converges uniformly to $f$.

Answer 1

HINT: The limit is $x^2$, and because $|f(x)-f_n(x)|<1/n$ it is the uniform convergence.

Answer 2

Note that if $n \geq 2$

$$\sup_{x}|f_n(x)-x^2| = \left|\frac1{n}-\frac1{n^2}\right| \rightarrow 0$$

and the sequence converges uniformly to $f(x) = x^2$.